The power dissipated by the resistor is

Using Ohm's Law

And changing the equation to be exclusively in terms of voltage and resistance

From this, it can be seen that doubling the resistance will halve the power dissipated if the voltage is kept constant.

To answer this question, we first need to determine what we're looking for that will allow any given light bulb to produce bright light. The answer is power. The more energy that is delivered to the light bulb within a given amount of time, the brighter the light will be. Let's go ahead and look at the equation for power in a circuit.

And since we're told in the question stem that the voltage source is identical in each circuit, we're looking for the circuit that has the largest current.

In order to find which circuit has the largest current, we'll need to invoke Ohm's law.

What this shows is that a higher current will occur in the circuit that has the lowest total resistance. Thus, we'll need to determine what the total resistance is in each of the three circuits.

In circuit A, we can see that there is only a single resistor. Thus, we can give this circuit's total resistance a value of .

In circuit B, we have two resistors that are connected in series. Remember that when resistors are connected in this way, the overall resistance of the circuit increases. We find the total resistance by summing all the resistors connected in series.

Therefore, we can give circuit B a total resistance of .

Now, let's look at circuit C. We can see that there are two resistors connected in parallel, and each of these are connected in series with a third resistor. To solve for the total resistance of this circuit, we first need to determine the equivalent resistance of the two resistors connected in parallel. Once we find that value, then we can take into account the third resistor connected in series.

To solve for the resistance of the two resistors connected in parallel, we have to remember that they add inversely.

Now that we've found the equivalent resistance for the two resistors connected in parallel, we can consider the third resistor connected in series.

Thus, we can give circuit C a value of .

Now that we've found the total resistance for each circuit, we can obtain our answer. Since circuit A has the lowest total resistance, it will also have the greatest current. Consequently, it will also have the greatest power delivered to its resistor (the light bulb), thus causing the light coming from that bulb to be the brightest.

In series, resistance adds conventionally.

Using Ohm's law:

Using definition of electric power:

Plugging in values:

Use Ohm's law: 

Converting  to 

Using definition of electric power:

Plugging in values:

Definition of electrical power:

Ohm's law:

Combining equations:

When adding resistors in series, they add directly

Thus, in this case, resistance would increase, and if the voltage is kept constant, the power dissipated by the circuit would decrease. Doubling the resistance would cut the power output in half.

Definition of electrical power:

Ohm's law:

Combining equations:

When adding resistors in parallel, they add inversely

Thus, in this case, resistance would decrease, and if the voltage is kept constant, the power dissipated by the circuit would increase. Adding identical resistors in parallel would halve the resistance, which would double the power dissipated.